Euler's Formula relates sinusoidal functions and a complex exponential by:
From this expression, we see that a complex exponential can provide an alternative representation for a real sinusoidal signal:
Complex exponentials are convenient for calculating multiplications or divisions of sinusoidal signals:
A complex signal
can be rewritten
.
We then define
as the complex amplitude (the time-invariant part), polar representation of amplitude and phase shift of the complex exponential signal. This term also represents a vector in the complex plane where and
.
We then have
. The
term has magnitude = 1 and can be interpreted as adding an incremental angle or phase shift to every time step .
In this way, is a phasor rotating in the complex plane with radian frequency
.